Chapter 2 heat transfer ppt

1

SHROFF S. R. ROTARY INSTITUTE OF CHEMICAL TECHNOLOGY (SRICT)

DEPARTMENT OF MECHANICAL ENGINEERING.

Subject: Applied Thermal and Hydraulic Engineering (ATHE)

Prepared by: Mr. Mitesh Gohil

Chapter 2. Heat Transfer

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Chapter 2. Heat Transfer

2.1 Modes of heat transfer

2.2 Conduction heat transfer

2.3 Convection heat transfer

2.4 Combine conduction and convection heat transfer

2.5 Heat exchangers

2.6 Radiation heat transfer

2.7 Summary

Outline

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2.1 Modes of heat transfer

The electrical engineering requires the knowledge of heat transfer for designing the

cooling system of motor, generator and transformers etc.

Modes of heat transfer

1) Conduction: In solid the heat energy transferred from one molecular to another

with conduction heat transfer via molecular lattice vibration and free electrons.

2) Convection: In fluid the heat energy is transferred with convection mode and it

is due to the movement of molecular.

3) Radiation: Heat energy due to emission of energy in the form of

electromagnetic waves by all bodies above 0 K temp. It does not required

medium, in fact most efficient in vacuum.

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𝑄 ∝ 𝐴𝑑𝑇

𝑑𝑥

𝑄 = −𝑘 𝐴𝑑𝑇

𝑑𝑥(Watt) (1)

𝑘 = − 𝑄

𝐴

𝑑𝑥

𝑑𝑇 W/m ºC

Fourier’s law of conduction:

“The rate of conduction heat transfer trough a plane layer is proportional to the

temperature difference across the layer and heat transfer area and inversely

proportional to the thickness of the layer.”

Thermal conductivity (k):

Thermal conductivity is rate of heat transfer through a unit thickness of the

material per unit area per unit temperature difference. It depends on the material

and temperature.

5

Heat conduction through plane wall (A Slab)

Fig. 2.1 Temperature

distribution in a plane wall

𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 = −𝑘 𝐴𝑑𝑇

𝑑𝑥

(Fourier’s law of

conduction)

𝑥=0

𝑥=𝐿

𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 𝑑𝑥 = −

𝑇1

𝑇2

𝑘𝐴 𝑑𝑇

𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 = 𝑘 𝐴𝑇1 − 𝑇2

𝐿

𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 =𝑇1 − 𝑇2

𝐿𝑘𝐴

(Rearranging above

equation)

𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 =𝑇1 − 𝑇2𝑅𝑤𝑎𝑙𝑙

(Thermal Resistance

Concept)

Thermal resistance, 𝑅𝑤𝑎𝑙𝑙 =𝐿

𝑘𝐴


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Fig. 2.2 Analogy between thermal and

electrical resistance concepts

Concept of thermal resistance


7

Heat conduction through hollow cylinder

Fig. 2.3 A long cylindrical pipe

𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 = −𝑘 𝐴𝑑𝑇

𝑑𝑟

(Fourier’s law of

conduction)

𝑟1

𝑟2 𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙

𝐴𝑑𝑟 = −

𝑇1

𝑇2

𝑘𝑑𝑇

𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 = 2𝜋𝐿𝑘𝑇1 − 𝑇2ln 𝑟2 𝑟1

𝐴 = 2𝜋𝑟𝐿

L= length of cylinder

𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 =𝑇1 − 𝑇2

ln𝑟2𝑟1

2𝜋𝐿𝑘

(Rearranging above

equation)

𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 =𝑇1 − 𝑇2

𝑅𝑐𝑦𝑙(Thermal Resistance

Concept)


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Thermal resistance (𝑅𝑐𝑦𝑙)

𝑅𝑐𝑦𝑙 =ln

𝑟2𝑟1

2𝜋𝐿𝑘


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Convection heat transfer

The convection is mode of heat transfer in which heat transfer by two

mechanisms random molecular motion (diffusion) and energy transfer

by bulk or macroscopic motion of the fluid.

Convection heat transfer may takes place between solid and fluid or

between two fluids or inside a fluid.


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Types:

Natural Convection

In natural (or free) convection the flow of fluid in induced by buoyancy

forces, which arise from density different caused by temperature variation in

fluid.

The layer of fluid which is in immediate vicinity of the plate gets heated by

conduction. This will increase the temperature of that layer and its density

will decrease. So, due to buoyancy force it will try to move upward and mix

with cold fluid.

Force Convection

Heat transfer which takes place in moving fluid is called as forced

convection. In force convection the heat transfer rate is higher as compare to

natural convection.


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𝑄𝑐𝑜𝑛𝑣. ∝ 𝐴 𝑇𝑠 − 𝑇𝑓

𝐴 = Surface area

𝑇𝑠 = Temperature of hot surface

𝑇𝑓 = Temperature of cold fluid

Newton’s law of cooling for convection heat transfer

“The rate of heat transfer per unit surface area from surface to fluid is

proportional to temperature difference between surface and fluid.”

𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓

h = heat transfer coefficient

h depends on surface geometry

and fluid properties.

𝑄𝑐𝑜𝑛𝑣. =𝑇𝑠 − 𝑇𝑓1ℎ 𝐴

(Rearranging above equation)

𝑄𝑐𝑜𝑛𝑣. =𝑇𝑠 − 𝑇𝑓

𝑅𝑐𝑜𝑛𝑣.Thermal Resistance Concept


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Thermal resistance (𝑅𝑐𝑜𝑛𝑣.)

𝑅𝑐𝑜𝑛𝑣. =1

ℎ 𝐴

Sr.

No.Heat transfer Case 𝑹𝒕𝒉

1 Conduction through Plane wall𝐿

𝑘𝐴

2Conduction through hollow

cylinder

ln𝑟2𝑟1

2𝜋𝐿𝑘

3 Convection1

ℎ 𝐴

Summary of thermal resistance

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Plane composite wall with both side fluids

Fig. 2.4 The temperature drop across a

layer is proportional to its thermal

resistance.

𝑄 =𝑇∞1 − 𝑇∞2

𝑅𝑡𝑜𝑡𝑎𝑙

𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑤𝑎𝑙𝑙 + 𝑅𝑐𝑜𝑛𝑣,2

𝑅𝑡𝑜𝑡𝑎𝑙 =1

ℎ1 𝐴+

𝐿

𝑘𝐴+

1

ℎ2 𝐴

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Multilayer Plane Walls with both side fluid

Fig. 2.5 The thermal resistance network for

heat transfer through a two-layer plane wall

subjected to convection on both sides.

𝑄 =𝑇∞1 − 𝑇∞2


𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑤𝑎𝑙𝑙,1 + 𝑅𝑤𝑎𝑙𝑙,2 + 𝑅𝑐𝑜𝑛𝑣,2


ℎ1 𝐴+

𝐿1𝑘1 𝐴

+𝐿2𝑘2 𝐴

+1

ℎ2 𝐴

15


Hollow composite cylinder with both side fluids

Fig. 2.6 The thermal resistance

network for heat transfer through a

three layered composite cylinder to

convection on both sides.

𝑄 =𝑇∞1 − 𝑇∞2


𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑐𝑦𝑙,1 + 𝑅𝑐𝑦𝑙,2 + 𝑅𝑐𝑦𝑙,3 + 𝑅𝑐𝑜𝑛𝑣,2


ℎ1 𝐴1+ln

𝑟2𝑟1

2𝜋𝐿𝑘1+ln

𝑟3𝑟2

2𝜋𝐿𝑘2+ln

𝑟4𝑟3

2𝜋𝐿𝑘3+

1

ℎ2 𝐴4

Where,𝐴1 = 2𝜋𝑟1𝐿 and 𝐴4 = 2𝜋𝑟4𝐿

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Critical Radius of Insulation

Fig. 2.7 An insulated cylindrical pipe

and the thermal resistance network

associated with it.

𝑄 =𝑇1 − 𝑇∞

𝑅𝑖𝑛𝑠 + 𝑅𝑐𝑜𝑛𝑣=

𝑇1 − 𝑇∞

ln𝑟2𝑟1

2𝜋𝐿𝑘+

1ℎ (2𝜋 𝑟2𝐿)

𝑄 =𝑇1 − 𝑇∞𝑅𝑡𝑜𝑡𝑎𝑙

Fig. 2.8 The variation of heat transfer

rate with the outer radius of the

insulation 𝑟2

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Above equation shows that if we decrease the outer radius 𝑟2, it decrease the

conductive resistance of wall(𝑅𝑖𝑛𝑠), but increase convective resistance (𝑅𝑐𝑜𝑛𝑣).

The heat transfer from the pipe may increase or decrease, depending on which

effect dominates.

The variation of 𝑄 with the outer radius of the insulation 𝑟2 is plotted in Fig.

The value of 𝑟2 at which 𝑄 reaches a maximum is determined from the

requirement that 𝑑 𝑄 𝑑𝑟2 = 0 (zero slope).

Performing the differentiation and solving for 𝑟2 yields the critical radius of

insulation for a cylindrical body to be

𝑟𝑐𝑟,𝑐𝑦𝑙 =𝑘

ℎ

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Fins and their application

We are interested to improve the heat dissipation from the surface of

motor, generator etc.

From Newton’s law of 𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓

So, it is possible to increase the heat dissipation (heat removal) by

increase in surface area or heat transfer co-efficient.

But h is constant and fix for air or according to fluid.

So, only way to increase the heat dissipation is to increase the surface area

by making extended surface or fins.

Cross sectional area of the extended surface which is called as profile of

fin may find in different form depending upon application.

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Table: 2.1 Fins type and its application

Sr.

No.Name Shape Application

1Straight

rectangular fins

IC Engine, Compressor,

Electric Motor2

Straight triangular

fins

3Straight parabolic

fins

4Circular fins of

rectangular profile

Heat Exchanger, Heat

Pipe, Refrigeration

Evaporator and Condenser

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5Pin fins of rectangular

profile

6Pin fins of triangular

profile

7Pin fins of parabolic

profile

8Pin fins of parabolic

profile (blunt tip)

Plate type heat

exchanger, fin

type heat

exchanger

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2.5 Heat exchangers

Definition

A Heat Exchanger may be defined as a Mechanical device which transfer the

heat from hot fluid to cold fluid, with maximum rate, with minimum

investment and with running cost.

Energy Balance: Enthalpy lost by hot fluid = enthalpy gained by cold fluid

(Assume negligible heat losses to surroundings)

Example:-

Condenser, Evaporator,

Boilers, super heater and reheaters,

Regenerative Heat Exchanger

Radiators of automobiles,

Oil coolers of heat engine,

Evaporator and condenser of

refrigeration system ,

Water and air coolers or heaters

Application:

Power production, process,

chemical, food and manufacturing

industries,

Electronics,

Waste heat recovery,

Refrigeration and air conditioning,

Space applications.

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2.5 Heat exchangers

Classification of Heat Exchangers

Recuperators / Regenerators

Recuperative: In these type of heat exchanger, the cold and hot fluid

simultaneously through the device and the heat is transferred through the wall

separating therm. (most commonly used)

Regenerative: In these type of heat exchanger, the hot and cold fluids

alternatively on the same surface. Thus same surface is subjected to periodic

heating and cooling.

Fig. 2.9 Recuperative and Regenerative

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2.5 Heat exchangers

According to Heat transfer mechanism:

single phase and two phase

Heaters or Cooler (sensible heat changes)

Condensers

In condensers, The hot fluid condense

(Gas to Liquid) at constant temperature

whilst temperature of cold fluid

gradually increases from inlet to outlet.

The hot fluid reject latent heat which is

absorbed by the cold fluid.

In case of condenser of thermal power

plant, the hot fluid is steam and the cold

fluid is cooling water which is gaining

the heat and rejecting the heat in cooling

towers.

Fig. 2.10 Temperature distribution

for condenser

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2.5 Heat exchangers

Evaporators

In evaporator cold fluid evaporates (Liquid

to Gas) at constant temperature whilst

temperature of hot fluid gradually

decreases from inlet to outlet.

In case of thermal power plants, the flue

gases (hot fluids) reject the heat and the

same is gain by the cold fluid (i.e. water) in

boilers.

In evaporator of refrigeration system, the

cold fluid (refrigerant) change its phase at

constant temperature by absorbing the heat

from hot fluid (air or water).

Fig. 2.10 Temperature distribution

for boiler (or evaporator)

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2.5 Heat exchangers

According to Transfer process:

Direct contact (mixing of fluid): In this type of heat exchanger, the two fluids

at different temperatures are come and mixed with direct contact. Example:

cooling tower

Indirect contact (No mixing) : In this type of heat exchanger, the two fluids at

different temperatures are exchange heat without direct contact. Example:

shell and tube heat exchanger.

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2.5 Heat exchangers

According to Flow arrangement:

Parallel flow/unidirectional flow:

The hot and cold fluids enter at

same end of the heat exchanger flow

through in same direction and leave

at other end. Since, low heat transfer

compare to counter flow generally

not used.

2.5 Heat exchangers

Counter flow/ Counter current heat

exchanger: The hot and cold fluid enter

at the opposite ends of heat exchanger

flow through in opposite direction and

leave at opposite ends.

Cross flow heat exchanger: The two

fluids flow at right angel to each other.

27

2.5 Heat exchangers

According to Geometry of construction;

Concentric tube type heat exchangers: It

is also called tubular or double pipe heat

exchanger.

Shell and tube type heat exchangers:

These are most commonly used for

heating, cooling, condensation or

evaporation where large heat transfer

required.

Fig. 2.13 Double pipe heat exchanger

Fig. 2.14 Shell-and-Tube Heat Exchanger28

2.5 Heat exchangers

Finned type: When an enhanced heat transfer rate is required, the

extended surfaces are used on one side of the heat exchanger.

Example: Liquid-Gas (Fins are always added on gas side)

Fig. 2.15 (a) Tube-Finned & (b) Plate -Finned plate heat exchanger29

2.5 Heat exchangers

Compact heat exchangers: These are special class of heat exchangers in

which the heat transfer surface area per unit volume is very large (>

700𝑚2/𝑚3)

Fig. 2.16 Plate and Frame Heat Exchanger 30

2.5 Heat exchangers

Overall heat transfer co-efficient

A heat exchanger typically involves two flowing fluids separated by a solid

wall. Heat is first transferred from the hot fluid to the wall by convection,

through the wall by conduction, and from the wall to the cold fluid again by

convection. Any radiation effects are usually included in the convection heat

transfer coefficients.

The thermal resistance network associated with this heat transfer process

involves two convection and one conduction resistances, as shown in Fig.

Here the subscripts i and o represent the inner and outer surfaces of the inner

tube.

For a double-pipe heat exchanger,

𝑄 =𝑇𝑖 − 𝑇𝑜𝑅𝑡𝑜𝑡𝑎𝑙

31

2.5 Heat exchangers

Fig. 2.17 Thermal resistance network

associated with heat transfer in a double-

pipe heat exchanger

Total thermal resistance,

𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,𝑖 + 𝑅𝑤𝑎𝑙𝑙 + 𝑅𝑐𝑜𝑛𝑣,𝑜


ℎ𝑖 𝐴𝑖+ln

𝑟𝑜𝑟𝑖

2𝜋𝐿𝑘+

1

ℎ𝑜 𝐴𝑜

In the analysis of heat exchangers, it is

convenient to combine all the thermal

resistances in the path of heat flow from the

hot fluid to the cold one into a single

resistance R, and to express the rate of heat

transfer between the two fluids as,

𝑄 =∆𝑇

𝑅𝑡𝑜𝑡𝑎𝑙= 𝑈𝐴𝑠∆𝑇 = 𝑈𝑖𝐴𝑖∆𝑇 = 𝑈𝑜𝐴𝑜∆𝑇

32

2.5 Heat exchangers

Where 𝐴𝑠 is the surface area and U is the overall heat transfer coefficient,

whose unit is W/𝑚2·K.

From above equation

1

𝑈 𝐴𝑠=

1

𝑈𝑖 𝐴𝑖=

1

𝑈𝑖 𝐴𝑖= 𝑅𝑡𝑜𝑡𝑎𝑙 =

1

ℎ𝑖 𝐴𝑖+ln

𝑟𝑜𝑟𝑖

2𝜋𝐿𝑘+

1

ℎ𝑜 𝐴𝑜

33

2.5 Heat exchangers

The Log Mean Temperature Difference Method (LMTD)

The temperature difference between the hot and cold fluids varies along the

heat exchanger.

It is convenient to have a mean temperature difference ∆ 𝑇𝑚 for use in the

relation

𝑄 = 𝑈 𝐴𝑠∆𝑇𝑚

LMTD for Counter flow

Consider the parallel-flow double-pipe heat exchanger.

Fig. 2.18 LMTD for counter flow 34

2.5 Heat exchangers

𝑑𝑄 = −𝑚ℎ𝑐𝑝ℎ𝑑𝑇ℎ = −𝑚𝑐𝑐𝑝𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃U=overall heat transfer co-efficient

𝜃 = 𝑇ℎ − 𝑇𝑐

𝑑𝑄 = −𝐶ℎ𝑑𝑇ℎ = −𝐶𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃(∵ 𝑚ℎ𝑐𝑝ℎ = 𝐶ℎ ;

𝑚𝑐𝑐𝑝𝑐 = 𝐶𝑐)…(1)

𝑄 = 𝐶ℎ(𝑇ℎ1 − 𝑇ℎ2) = 𝐶𝑐(𝑇𝑐1 − 𝑇𝑐2) …(2)

Energy balance (counter flow) on element shown

Energy balance from inlet to outlet

35

Fig. 2.18 LMTD for counter flow

2.5 Heat exchangers

From equation (1) 𝑑𝑇ℎ − 𝑑𝑇𝑐

𝑑𝑇ℎ − 𝑑𝑇𝑐 = 𝑑𝑄1

𝐶𝑐−

1

𝐶ℎ

𝑑𝜃 = 𝑈 𝑑𝐴 𝜃1

𝐶𝑐−

1

𝐶ℎFrom eq. (1)

1

2𝑑𝜃

𝜃=

1

2

𝑈𝑑𝐴1

𝐶𝑐−

1

𝐶ℎ

Integrate from

end 1 to 2

ln𝜃2𝜃1

= 𝑈𝐴1

𝐶𝑐−

1

𝐶ℎ

ln𝜃2𝜃1

= 𝑈𝐴(𝑇𝑐1 − 𝑇𝑐2)

𝑄−(𝑇ℎ1 − 𝑇ℎ2)

𝑄From eq. (2)


36

2.5 Heat exchangers

ln𝜃2𝜃1

= 𝑈𝐴(𝑇ℎ2 − 𝑇𝑐2)

𝑄−(𝑇ℎ1 − 𝑇𝑐1)

𝑄(Rearranging terms)

ln∆𝑇2∆𝑇1

= −𝑈𝐴∆𝑇2𝑄

−∆𝑇1𝑄

∆𝑇1= (𝑇ℎ1 − 𝑇𝑐1)

∆𝑇2=(𝑇ℎ2 − 𝑇𝑐2)

𝑄 = 𝑈𝐴∆𝑇2 − ∆𝑇1

ln∆𝑇2∆𝑇1

𝑄 = 𝑈𝐴 𝑇𝑙𝑚


37

2.5 Heat exchangers

LMTD for Parallel flow

Consider the parallel-flow double-pipe heat exchanger.

Fig. 2.19 LMTD for parallel flow

𝑑𝑄 = −𝑚ℎ𝑐𝑝ℎ𝑑𝑇ℎ = 𝑚𝑐𝑐𝑝𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃U=overall heat transfer co-efficient

𝜃 = 𝑇ℎ − 𝑇𝑐

𝑑𝑄 = −𝐶ℎ𝑑𝑇ℎ = 𝐶𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃(∵ 𝑚ℎ𝑐𝑝ℎ = 𝐶ℎ ;

𝑚𝑐𝑐𝑝𝑐 = 𝐶𝑐)…(1)

𝑄 = 𝐶ℎ(𝑇ℎ1 − 𝑇ℎ2) = 𝐶𝑐(𝑇𝑐2 − 𝑇𝑐1) …(2)

Energy balance (parallel flow) on element shown

Energy balance from inlet to outlet38

2.5 Heat exchangers

𝑑𝑇ℎ − 𝑑𝑇𝑐 = −𝑑𝑄1

𝐶𝑐+

1

𝐶ℎ

𝑑𝜃 = −𝑈 𝑑𝐴 𝜃1

𝐶𝑐+

1

𝐶ℎFrom eq. (1)

1

2𝑑𝜃

𝜃= −

1

2

𝑈𝑑𝐴1

𝐶𝑐+

1

𝐶ℎ

Integrate from end

1 to 2

ln𝜃2𝜃1

= −𝑈𝐴1

𝐶𝑐+

1

𝐶ℎ

ln𝜃2𝜃1

= −𝑈𝐴(𝑇𝑐2 − 𝑇𝑐1)

𝑄+(𝑇ℎ1 − 𝑇ℎ2)

𝑄From eq. (2)

From equation (1) 𝑑𝑇ℎ − 𝑑𝑇𝑐

Fig. 2.19 LMTD for parallel flow

39

2.5 Heat exchangers

Fig. 2.19 LMTD for parallel flowln𝜃2𝜃1

= 𝑈𝐴(𝑇𝑐1 − 𝑇𝑐2)

𝑄−(𝑇ℎ1 − 𝑇ℎ2)

𝑄

ln𝜃2𝜃1

= 𝑈𝐴(𝑇ℎ2 − 𝑇𝑐2)

𝑄−(𝑇ℎ1 − 𝑇𝑐1)

𝑄(Rearranging terms)

ln∆𝑇2∆𝑇1

= 𝑈𝐴∆𝑇2𝑄

−∆𝑇1𝑄

∆𝑇1= (𝑇ℎ1 − 𝑇𝑐1)

∆𝑇2=(𝑇ℎ2 − 𝑇𝑐2)

𝑄 = 𝑈𝐴∆𝑇2 − ∆𝑇1

ln∆𝑇2∆𝑇1

𝑄 = 𝑈𝐴 𝑇𝑙𝑚𝑡𝑑 40

2.5 Heat exchangers

Temperature distribution and ∆𝑇1&∆𝑇2 for condenser and Evaporator

(a) Condenser (b) Evaporator or Boiler

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Radiation heat transfer: “The transfer of heat across a system boundary by

means of electro-magnetic waves which is caused solely by a temperature

difference.”

Does not required medium and occurs most effectively in vacuum and occurs

above 0 K temperature.

Radiation heat transfer occurs by electro-magnetic waves or quanta of energy

called photons.

The photons are propagated through space as ray (trance with light speed)

which unchanged frequency.

Spectrum of electro-magnetic radiation contain γ-ray, X-ray, UV, Infra-red,

thermal radiation (Include Visible), and micro waves etc.

Out of all above radiation only thermal radiation (λ=10−1𝜇𝑚 𝑡𝑜 102𝜇𝑚) is of

our interest.

Example: Solar, electric bulb, furnaces, nuclear explosions etc.

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𝐸𝑏 = 𝜎𝐴𝑇4Unit of 𝐸𝑏 𝑖𝑠 Watt &

𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4

= Stefan Boltzmann constant

𝐸 =

0

𝜆

𝐸𝜆 𝑑𝜆 W/𝑚2

2) Monochromatic (Spectral) emissive power

The amount of radiation energy emitted from a surface at a given

temperature varies with variation in wavelength (λ).

Monochromatic emissive power is defined as the amount of radiant energy

emitted by a surface at given temperature per unit area, per unit time.

Surface Emission Properties:

1) Total emissive power (E)

It is the total amount of radiation emitted by a black body per unit area and time.

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𝐸 = 𝜀𝜎𝐴𝑇4Unit of 𝐸 𝑖𝑠 Watt &

ε = emissivity of the material

𝜀 =𝐸

𝐸𝑏

3) Emission form a real surface emissivity

4) Emissivity (ε) is defined as the ability of the surface of a body to radiate

heat.

It is the ratio of emissive power of any body to black body of equal

temperature.

It may varies with temperature and wavelength.

5) Intensity of radiation or Irradiation

It is defined as the total incident radiation on a surface from all the

directions per unit area per unit time expressed in W/𝑚2.

6) Radiosity:

It is defined as the total radiant energy leaving from the surface from all the

directions per unit area per unit time expressed in W/𝑚2.

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Concepts of different bodies

𝐺 = 𝐺𝑟 + 𝐺𝑎 + 𝐺𝑡

1 =𝐺𝑟𝐺+𝐺𝑎𝐺+𝐺𝑡𝐺

1 = 𝜌 + 𝛼 + 𝜏

(1) Black Body: Perfect Absorbing body with unit absorptivity (𝛼 = 1) is called as black body.

(2) White Body: Perfect reflective body with unit reflectivity (𝜌 = 1)is called as White body.

(3) Opaque Body: Perfect transparent body with zero transmissivity(𝜏 = 0) is called as Opaque

body.

(4) Grey Body

If the radiative properties, 𝛼, 𝜌, 𝜏 of a body are assumed to be uniform over the entire

wave length spectrum, then such a body is called Grey body.

One whose absorptivity does not vary with temperature and wavelength of the incident

radiation (𝛼 = (𝛼)𝜆= 𝑐𝑜𝑛𝑠𝑡.)

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Laws of radiation

𝐸𝑏 = 𝜎𝐴𝑇4Unit of 𝐸𝑏 𝑖𝑠 Watt &

𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4


Stefan Boltzmann

The Stefan Boltzmann states that the emissive power of black body is directly

proportional to the fourth power of its absolute temperature.

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Kirchhoff’s law

This law states that at any temperature the ratio of total emissive power (E) to

the total absorptivity (𝛼) is constant for all substance which are in thermal

equilibrium with environment.𝐸

𝛼= 𝑐𝑜𝑛𝑠𝑡.

𝐸

𝛼= 𝐸𝑏

Now we know, 𝐸 = 𝜀𝜎𝐴𝑇4 & 𝐸𝑏 = 𝜎𝐴𝑇4

𝜀𝜎𝐴𝑇4

𝛼= 𝜎𝐴𝑇4

𝜀 = 𝛼

Emissivity (ε) of a body is equal to Absorptivity (𝛼) of the body in thermal

equilibrium with environment.

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2.7 Summary

1) Conduction:(Fourier Law)

2) Convection: (Newton’s law of cooling)

3) Radiation:

𝑄 = −𝑘 𝐴𝑑𝑇

𝑑𝑥

(Watt)

k = thermal conductivity

k depends on material

𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓

h = heat transfer coefficient

h depends on surface geometry

and fluid properties.

𝐸 = 𝜀𝜎𝐴𝑇4

Unit of 𝐸 𝑖𝑠 Watt &

ε = emissivity of the material

𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4


Modes of heat transfer

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